Exploring and exploiting new representations for multivariate extremes

点击翻译按钮获取中文摘要

In all aspects of life, there is a need to proportionally mitigate against the risk posed by rare but potentially catastrophic events. For example, we protect ourselves from flooding through the construction of defences: in doing so, we balance risk and cost by building these high enough such that the probability of them being breached over their lifetime is small, but not so high that money is wasted on eliminating infinitesimal risks. Usually, we will be trying to protect against extreme events that are larger than we have ever observed, meaning that direct estimation of the probability of breach from existing data is impossible. Extreme value theory is the mathematically-justified approach for tackling this problem: we can learn from extremes we have seen to estimate probabilities of events not yet witnessed.Events that cause the most impact are often multivariate or spatial in nature. For example, damage to a structure may occur during a period of high winds, but that damage could be far more costly when accompanied by high rainfall. Equally, a large loss in a single element of a financial portfolio is less disastrous than multiple losses across the board. In order to understand the risks posed by such phenomena, we need tools for modelling the dependence between processes at extreme levels. To date there are a variety of methods available, each based on different underlying assumptions, and the extent to which these represent good statistical models rests strongly on the unknown underlying dependence. In this work we will exploit novel representations and recently-uncovered links between these methods to unify these disparate methodologies and provide a single, reliable strategy for modelling multivariate extremes.